The Mathematics of Financial Modelingand Investment Management

6-ConceptsProbability Page 190 Wednesday, February 4, 2004 3:00 PM

190 The Mathematics of Financial Modeling and Investment Management

of pointwise convergence. A sequence of random variables Xi, i ≥1 on

(Ω,ℑ,P), is said to converge almost surely to a random variable X,

denoted

a.s.

Xi →X

if the following relationship holds:

P{ω: lim Xi ()ω = X ()ω}= 1

i → ∞

In other words, a sequence of random variables converges almost surely

to a random variable X if the sequence of real numbers Xi(ω) converges

to X(ω) for all ωexcept a set of measure zero.

A sequence of random variables Xi, i ≥1 on (Ω,ℑ,P), is said to con-

verge in mean of order p to a random variable X if

lim EX[ ()– X ω p

i → ∞ i ω ()]=^0

provided that all expectations exist. Convergence in mean of order one

and two are called convergence in mean and convergence in mean

square, respectively.

A weaker concept of convergence is that of convergence in probabil-

ity. A sequence of random variables Xi, i ≥1 on (Ω,ℑ,P), is said to con-

verge in probability to a random variable X, denoted

P

Xi →X

if the following relationship holds:

lim P{ω: Xi ()ω – X ()ω ≤ ε}= 1 , ∀ε> 0

i → ∞

It can be demonstrated that if a sequence converges almost surely

then it also convergences in probability while the converse is not gener-

ally true. It can also be demonstrated that if a sequence converges in

mean of order p > 0, then it also convergences in probability while the

converse is not generally true.

A sequence of random variables Xi, i ≥1 on (Ω,ℑ,P) with distribution

functions FXi is said to converge in distribution to a random variable X

with distribution function FX, denoted